Best bounds for the Hilbert transform on \(L^ p(\mathbb{R}^ 1)\) (Q1372884)
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scientific article; zbMATH DE number 1082974
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Best bounds for the Hilbert transform on \(L^ p(\mathbb{R}^ 1)\) |
scientific article; zbMATH DE number 1082974 |
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Best bounds for the Hilbert transform on \(L^ p(\mathbb{R}^ 1)\) (English)
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4 November 1997
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In this note the author gives a short proof (which avoids the use of conjugate function on the circle) of the following known best bounds result: The Hilbert transform \(H\), \[ Hf(x)=\text{p.v. }\Biggl({1\over\pi}\Biggr) \int_\mathbb{R} {f(x-t)\over t} dt, \] maps \(L^p(\mathbb{R})\) into itself with bound \(\tan(\pi/2p)\) for \(1<p\leq 2\) and \(\text{cot}(\pi/2p)\) for \(2\leq p<\infty\).
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\(L^ p\) spaces
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subharmonic function
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Hilbert transform
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